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Exponential function

  1. Feb 13, 2006 #1
    Hi,
    I'm working on this:
    Given that [tex]\lim_{n \to \infty} \psi(n)=0[/tex] and that b and c do not depend upon n, prove that:
    [tex]\lim_{n\to \infty}\left[ 1+\frac{b}{n} +\frac{\psi(n)}{n}\right]^{cn} = \lim_{n\to\infty} \left(1+\frac{b}{n}\right)^{cn}=e^{bc}[/tex]
    So far, I've taken the natural log of both sides, moved the cn into the bottom and applied L'hopitals rule. I get:
    [tex]\lim_{n\to\infty}\frac{\frac{1}{1+\frac{b}{n} +\frac{\psi(n)}{n}}\left(\frac{-b}{n^2} + \frac{\psi'(n)}{n}-\frac{\psi(n)}{n^2}\right)}} {\frac{-1}{c n^2}}}[/tex][tex]=\lim_{n\to \infty}bc[/tex]
    which breaks down to:
    [tex]\lim_{n\to\infty}\frac{1}{1+\frac{b}{n}+\frac{\psi(n)}{n}}\left(-cn\psi'(n)-c\psi(n)+bc)\right)[/tex]
    If the limit of a function goes to zero, how do we prove that it's derivative goes to zero?
    I'm not sure where to go now, because I don't know what to do with [tex]\psi'(n)[/tex] how can I prove that it's zero? If it IS zero, then the whole thing falls out nicely.
    Thanks,
    CC
     
    Last edited: Feb 13, 2006
  2. jcsd
  3. Feb 13, 2006 #2

    quasar987

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    There is some problem with your LaTeX.

    Is psi a wave function by any chance?
     
  4. Feb 13, 2006 #3
    yea, I tried to fix the latex a little bit. For some reason, it's not centering my fraction on the bottom.That whole mess is over [tex]\frac{-1}{c^2 n^2}[/tex]. No, the [tex]\psi(n)[/tex] isn't a wave function. This is for a statistics class. I think the only thing I'm supposed to worry about is that it goes to 0 as n goes to infinity. I also forrgot my limit in front of my giant fraction. I will try and fix it.
    CC
     
  5. Feb 13, 2006 #4
    There. It's a little better. I hope you can tell what I mean. I can't figure out why the bottom fraction is way over on the left like that. That WHOLE thing on the left equals [tex]\lim_{n\to\infty}bc[/tex]...which is just bc....anyway, any pointers will be appreciated. I am stuck stuck stuck.
    CC
     
  6. Dec 29, 2009 #5
    Well I know this post is 3 years old so you likely no longer need the answer but who knows, maybe someone will make some use of it.

    I will try and type it with LaTeX code but I am not positive it will show as intended as this is my first use of these forums.

    The basic principle used here will be the well known exponential convergence which is

    [tex]\lim_{n \to \infty} (1 + \frac{x}{n})^{n} = e^{x}[/tex]

    Now looking at your problem. Since c is independant from n, we have

    [tex]\lim_{n\to \infty}\left[ 1+\frac{b}{n} +\frac{\psi(n)}{n}\right]^{cn} & = & {\left[ \lim_{n\to \infty}\left[ 1+\frac{b}{n} +\frac{\psi(n)}{n}\right]^{n} \right]}^{c} [/tex]
    [tex]= {\left[ \lim_{n\to \infty}\left[ 1+\frac{b+\psi(n)}{n}\right]^{n} \right]}^{c}[/tex]
    [tex]= {\left[ \lim_{n\to \infty} e^{b+\psi(n)} \right]}^{c}[/tex]
    [tex]= {\left[ e^{b} \times \lim_{n\to \infty} e^{\psi(n)} \right]}^{c}[/tex]
    [tex]= {\left[ e^{b}\times e^{0} \right]}^{c}[/tex]
    [tex]= e^{bc}[/tex]



    Vincent
    Graduate math student
     
    Last edited: Dec 29, 2009
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